Numerous analog mixed-signal integrated circuits contain continuous-time filters based on elementary integrator cells that generally include a tunable transconductance amplifier (Gi) and a capacitor (Ci). Elementary integrator cells (Gi, Ci) are often building blocks of more complex circuits.
An exemplary integrator is shown in FIG. 1. The meaning of each block is summarized in the following table:
Giinput transconductance amplifierGmitransconductance amplifier matchedwith GiAioerror amplifierDACDigital-to-Analog ConverterVrreference voltageViinput voltageVooutput voltageRioutput resistance of thetransconductance amplifierCifilter capacitorRrreference resistorVgtransconductance control signal
The resistance Ri may be the finite output impedance of the transconductance amplifier Gi (due to the finite output impedance of the electronic components constituting the transconductance amplifier) or a resistive element (a resistor or an active device adapted to behave as a resistor) connected at the output of the transconductance amplifier, or a combination of the two.
The transfer function of the integrator of FIG. 1 is:
                              F          ⁡                      (            f            )                          =                                            G              i                        ⁢                          R              i                                            1            +                          i              ⁢                                                          ⁢              2              ⁢              π              ⁢                                                          ⁢                              f                ·                                  C                  i                                            ⁢                              R                i                                                                        (        1        )            
The depicted circuit (Gi-Ci) nominally behaves like an integrator having its 0 dB crossing frequency at fi=Gi/2πCi, wherein the transconductance gain Gi may be fixed in a very large range of values.
The error amplifier Aio adjusts the gain of the transconductance amplifier Gi and of the matched transconductance amplifier Gmi in order to nullify the difference between the reference current output by the DAC and the current Gmi*Vr of the transconductance amplifier Gmi:
                              G          mi                =                              G            DAC                                R            r                                              (        2        )            wherein GDAC is the programmable current gain of the digital-to-analog converter. Using the reference resistor Rr that may be an external resistor, or a resistor with a minimal dependence from process spread and/or supply voltage and temperature fluctuations, the transconductance gain Gmi is adjusted by the control loop for compensating such possible fluctuations, and is tunable by adjusting the gain of the DAC.
When the amplifiers Gi and Gmi are matched, their transconductance may substantially vary according to the same law. However, the transconductance gain Gi and the resistance Ri, and thus also the voltage gain (Gi Ri) of the integrator cell are influenced by process spread, supply voltage, and temperature fluctuations (PVT). This makes the phase of the transfer function of the integrator cell at the crossing frequency fi different from 90°, as it would be in an ideal integrator, and causes a PVT-dependent phase shift. This phase shift may affect the frequency response of filters based on integrator cells, especially if they function at high frequencies and are tunable over a wide frequency range.